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Value at Risk and Expected Shortfall: Traditional Measures and Extreme Value Theory Enhancements

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Value at Risk and Expected Shortfall: Traditional Measures and Extreme Value Theory Enhancements VALUE AT RISK AND EXPECTED SHORTFALL: TRADITIONAL MEASURES AND EXTREME VALUE THEORY ENHANCEMENTS WITH A SOUTH AFRICAN MARKET APPLICATION by Anelda Dicks Assignment presented at the University of Stellenbosch in partial fulfilment of the requirements for the degree of Masters of Commerce Mast ers of C ommerc e Department of Statistics and Actuarial Science University of Stellenbosch Private Bag X1, 7602 Matieland, South Africa Supervisors: Prof. W.J. Conradie Prof. T. de Wet December 2013 i Stellenbosch University http://scholar.sun.ac.za Copyright © 2013 University of Stellenbosch All rights reserved. ii Stellenbosch University http://scholar.sun.ac.za Declaration I, the undersigned, hereby declare that the work contained in this assignment is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree. Signature: ………………………………. A. Dicks Date: …………………………………….. iii Stellenbosch University http://scholar.sun.ac.za Abstract Accurate estimation of Value at Risk (VaR) and Expected Shortfall (ES) is critical in the management of extreme market risks. These risks occur with small probability, but the financial impacts could be large. Traditional models to estimate VaR and ES are investigated. Following usual practice, 99% 10 day VaR and ES measures are calculated. A comprehensive theoretical background is first provided and then the models are applied to the Africa Financials Index from 29/01/1996 to 30/04/2013. The models considered include independent, identically distributed (i.i.d.) models and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) stochastic volatility models. Extreme Value Theory (EVT) models that focus especially on extreme market returns are also investigated. For this, the Peaks Over Threshold (POT) approach to EVT is followed. For the calculation of VaR, various scaling methods from one day to ten days are considered and their performance evaluated. The GARCH models fail to converge during periods of extreme returns. During these periods, EVT forecast results may be used. As a novel approach, this study considers the augmentation of the GARCH models with EVT forecasts. The two-step procedure of pre- filtering with a GARCH model and then applying EVT, as suggested by McNeil (1999), is also investigated. This study identifies some of the practical issues in model fitting. It is shown that no single forecasting model is universally optimal and the choice will depend on the nature of the data. For this data series, the best approach was to augment the GARCH stochastic volatility models with EVT forecasts during periods where the first do not converge. Model performance is judged by the actual number of VaR and ES violations compared to the expected number. The expected number is taken as the number of return observations over the entire sample period, multiplied by 0.01 for 99% VaR and ES calculations. iv Stellenbosch University http://scholar.sun.ac.za Opsomming Akkurate beraming van Waarde op Risiko (Value at Risk) en Verwagte Tekort (Expected Shortfall) is krities vir die bestuur van ekstreme mark risiko’s. Hierdie risiko’s kom met klein waarskynlikheid voor, maar die finansiële impakte is potensieel groot. Tradisionele modelle om Waarde op Risiko en Verwagte Tekort te beraam, word ondersoek. In ooreenstemming met die algemene praktyk, word 99% 10 dag maatstawwe bereken. ‘n Omvattende teoretiese agtergrond word eers gegee en daarna word die modelle toegepas op die Africa Financials Index vanaf 29/01/1996 tot 30/04/2013. Die modelle wat oorweeg word sluit onafhanklike, identies verdeelde modelle en Veralgemeende Auto-regressiewe Voorwaardelike Heteroskedastiese (GARCH) stogastiese volatiliteitsmodelle in. Ekstreemwaarde Teorie modelle, wat spesifiek op ekstreme mark opbrengste fokus, word ook ondersoek. In hierdie verband word die Peaks Over Threshold (POT) benadering tot Ekstreemwaarde Teorie gevolg. Vir die berekening van Waarde op Risiko word verskillende skaleringsmetodes van een dag na tien dae oorweeg en die prestasie van elk word ge- evalueer. Die GARCH modelle konvergeer nie gedurende tydperke van ekstreme opbrengste nie. Gedurende hierdie tydperke, kan Ekstreemwaarde Teorie modelle gebruik word. As ‘n nuwe benadering oorweeg hierdie studie die aanvulling van die GARCH modelle met Ekstreemwaarde Teorie vooruitskattings. Die sogenaamde twee-stap prosedure wat voor-af filtrering met ‘n GARCH model behels, gevolg deur die toepassing van Ekstreemwaarde Teorie (soos voorgestel deur McNeil, 1999), word ook ondersoek. Hierdie studie identifiseer sommige van die praktiese probleme in model passing. Daar word gewys dat geen enkele vooruistkattingsmodel universeel optimaal is nie en die keuse van die model hang af van die aard van die data. Die beste benadering vir die data reeks wat in hierdie studie gebruik word, was om die GARCH stogastiese volatiliteitsmodelle met Ekstreemwaarde Teorie vooruitskattings aan te vul waar die voorafgenoemde nie konvergeer nie. Die prestasie van die modelle word beoordeel deur die werklike aantal Waarde op Risiko en Verwagte Tekort oortredings met die verwagte aantal te vergelyk. Die verwagte aantal word geneem as die aantal obrengste waargeneem oor die hele steekproefperiode, vermenigvuldig met 0.01 vir die 99% Waarde op Risiko en Verwagte Tekort berekeninge. v Stellenbosch University http://scholar.sun.ac.za Acknowledgements The author thanks the following people for their contribution towards this study: • Prof. Tertuis de Wet, my co-study leader for theoretical assistance and proof reading as well as for moral support and encouragement throughout the research period. Thank you for unforgettable study years and valuable life insights. • Prof. Willie Conradie, co-study leader for research suggestions, meticulous proof reading and guidance during my research. Your passion about financial risk management is inspiring. Thank you. • Prof. N.J. le Roux for advice on plots in R. • My family for their patience and emotional support. • My friends for all their interest and for making this journey a pleasant one. vi Stellenbosch University http://scholar.sun.ac.za Contents Declaration iii Abstract iv Opsomming v Acknowledgements vi Contents vii List of Abbreviations xii List of Notation xiii 1. Introduction 1 2. Brief Literature Review of Extreme Value Theory Estimation Approaches 3 2.1. General results 3 2.2. Alternatives to the GPD 5 2.3. Empirical quantile approach 6 2.4. Semi-nonparametric EVT approach 6 vii Stellenbosch University http://scholar.sun.ac.za 2.5. Alternative to dynamic back-testing 7 3. Traditional VaR and ES measures 8 3.1. Basic Definitions 8 3.2. Comparison of VaR and ES 10 3.3. Models for independent, identically distributed returns 11 3.3.1. Normal model 11 3.3.2. Student t model 13 3.3.3. The problem with the i.i.d. assumption 13 3.4. The GARCH conditional model 14 3.4.1. Model description 14 3.4.2. Maximum likelihood estimation 16 3.4.2.1. Normal model 17 3.4.2.2. Student t model 18 3.4.3. Forecasts and long term volatility 20 3.4.3.1. Long term volatility 21 3.4.3.2. One day ahead forecasts 21 3.4.3.3. Term structure of volatility forecasts 23 3.4.3.4. Convergence of forecasts to long term volatility 23 3.5. Asymmetric GARCH conditional models 24 3.5.1. The rationale for asymmetric models 24 3.5.2. A-GARCH and GJR-GARCH 25 3.5.2.1. Model definition, long term variance and forecasts 25 3.5.2.2. Estimation 26 viii Stellenbosch University http://scholar.sun.ac.za 3.5.3. E-GARCH 26 3.5.3.1. Model definition 26 3.5.3.2. Properties 27 3.5.3.3. Long term variance 28 3.5.3.4. Estimation 29 3.5.3.5. One day forecasts 30 3.6. Summary 32 4 Traditional EVT 33 4.1 Introduction 33 4.2 Method of Block Maxima 34 4.3 Peaks over Threshold (POT) approach 37 4.3.1 Theoretical motivation for using the GPD 38 4.3.2 Generalised Pareto Distribution (GPD) 39 4.3.2.1 Parameter estimation 39 4.4 Fréchet class of distributions 42 4.4.1 Basic Definitions 43 4.4.2 The extreme value index 44 4.4.3 Threshold selection and exploratory data analysis 46 4.4.3.1. Quantile-Quantile (QQ) plots 47 4.4.3.2. Hill plots 48 4.4.3.3. Mean excess plots 48 4.5 Quantile estimation 49 4.5.1 First order estimation 50 ix Stellenbosch University http://scholar.sun.ac.za 4.5.1.1 The Weissman estimator 50 4.5.1.2 The probability approach estimator 52 4.5.2 Second order refinements 54 4.5.2.1 Quantile view 54 4.5.2.2 Probability view 57 4.6 Derivation of VaR and ES using the POT approach 60 4.6.1 Confidence intervals for VaR and ES 62 4.7 Limitations of EVT 65 4.8 Scaling of VaR and ES measures 66 4.8.1 The square root of time rule 66 4.8.2 The alpha root rule 66 4.8.3 Empirically established scaling laws 67 4.9 Summary 68 5. Combining EVT with stochastic volatility models 70 5.1. Two step procedure 70 5.3. h-day time horizons 72 6. Practical issues in model fitting 73 7. The data 75 7.1 Description of the data 75 x Stellenbosch University http://scholar.sun.ac.za 8. Analysis of the Africa Financials Index Data 81 8.1. Schematic Overview 81 8.2. Independent identically distributed models 82 8.3. GARCH models 85 8.3.1. Estimation results 85 8.3.2. Analysis of the absolute exceedances 89 8.4. Extreme Value Analysis 93 8.4.1. Exploratory analysis for the first data window 93 8.4.2. VaR and ES over the whole data series 105 8.4.2.1. One day VaR and ES estimates 107 8.4.2.2.
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